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In a logistic regression a positive/negative beta tells you that the direction that variable works. Is there anything in the Random Forest variable importance measures that indicates the direction of the variable?

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One possibility (without variable importance) would be to display partial dependence plots, which show you the connection between the variable and (one) predicted class. See partialPlot in randomForest package in R for more information.

You can also use the Partial prediction options in mlr http://mlr-org.github.io/mlr-tutorial/devel/html/partial_prediction/index.html

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